On some Frobenius groups with the same prime graph as the almost simple group ${ {\bf PGL(2,49)}}$
الموضوعات :
A. Mahmoudifar
1
(Department of Mathematics, Tehran North Branch, Islamic Azad University, Tehran, Iran)
الکلمات المفتاحية: prime graph, Almost simple group, element order, Frobenius group,
ملخص المقالة :
The prime graph of a finite group $G$ is denoted by$\Gamma(G)$ whose vertex set is $\pi(G)$ and two distinct primes $p$ and $q$ are adjacent in $\Gamma(G)$, whenever $G$ contains an element with order $pq$. We say that $G$ is unrecognizable by prime graph if there is a finite group $H$ with $\Gamma(H)=\Gamma(G)$, in while $H\not\cong G$. In this paper, we consider finite groups with the same prime graph as the almost simple group $\textrm{PGL}(2,49)$. Moreover, we construct some Frobenius groupswhose prime graphs coincide with $\Gamma(\textrm{PGL}(2,49))$, in particular, we get that $\textrm{PGL}(2,49)$ is unrecognizable by its prime graph.
[1] Z. Akhlaghi, M. Khatami, B. Khosravi, Characterization by prime graph of PGL(2, pk) where p and k are odd, International Journal of Algebra and Computation. 20 (2010), 847-873.
[2] A. A. Buturlakin, Spectra of Finite Symplectic and Orthogonal Groups, Siberian Advances in Mathematics. 21 (2011), 176-210.
[3] G. Y. Chen, V. D. Mazurov, W. J. Shi, A. V. Vasil’ev, A. Kh. Zhurtov, Recognition of the finite almost simple groups PGL2(q) by their spectrum, J. Group Theory. 10 (2007), 71-85.
[4] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, R. A. Wilson, Atlas of Finite Groups, Oxford University Press, Oxford, 1985.
[5] M.A. Grechkoseeva, On element orders in covers of finite simple classical groups, J. Algebra. 339 (2011), 304-319.
[6] D. Gorenstein, Finite Groups, Harper and Row, New York, 1968.
[7] M. Hagie, The prime graph of a sporadic simple group, Comm. Algebra. 31 (2003), 4405-4424.
[8] G. Higman, Finite groups in which every element has prime power order, J. London Math. Soc. 32 (1957), 335-342.
[9] M. Khatami, B. Khosravi, Z. Akhlaghi, NCF-distinguishability by prime graph of PGL(2, p), where p is a prime, Rocky Mountain J. Math. 41 (2011), 1523-1545.
[10] B. Khosravi, n-Recognition by prime graph of the simple group P SL(2, q), J. Algebra. Appl. 7 (2008), 735-748.
[11] B. Khosravi, 2-Recognizability of P SL(2, p2) by the prime graph, Siberian Math. J. 49 (2008), 749-757.
[12] B. Khosravi, B. Khosravi, B. Khosravi, On the prime graph of PSL(2, p) where p > 3 is a prime number, Acta. Math. Hungarica. 116 (2007), 295-307.
[13] R. Kogani-Moghadam, A. R. Moghaddamfar, Groups with the same order and degree pattern, Sci. China. Math. 55 (2012), 701-720.
[14] A. S. Kondrat’ev, Prime graph components of finite simple groups, Math. USSR-SB. 67 (1990), 235-247.
[15] A. Mahmoudifar, B. Khosravi, On quasirecognition by prime graph of the simple groups $A^+_n(p)$ and $A^−_n(p)$, J. Algebra. Appl. 14 (2015), 12 pages.
[16] A. Mahmoudifar, B. Khosravi, On the characterization of alternating groups by order and prime graph, Sib. Math. J. 56 (2015), 125-131.
[17] V. D. Mazurov, Characterizations of finite groups by sets of their element orders, Algebra Logic. 36 (1997), 23-32.
[18] A. R. Moghaddamfar, W. J. Shi, The number of finite groups whose element orders is given, Beitrage Algebra Geom. 47 (2006), 463-479.
[19] D. S. Passman, Permutation groups, W. A. Bengamin, New York, 1968.
[20] A. V. Zavarnitsine, Fixed points of large prime-order elements in the equicharacteristic action of linear and unitary groups, Sib. Electron. Math. Rep. 8 (2011), 333-340.
